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Formation and Evolution of Planetary Systems in Presence of Highly Inclined Stellar Perturbers Konstantin Batygin1,2, Alessandro Morbidelli1, and Kleomenis Tsiganis3

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Formation and Evolution of Planetary Systems in Presence of Highly Inclined Stellar Perturbers Konstantin Batygin1,2, Alessandro Morbidelli1, and Kleomenis Tsiganis3 Astronomy & Astrophysics manuscript no. ms c ESO 2018 October 29, 2018 Formation and Evolution of Planetary Systems in Presence of Highly Inclined Stellar Perturbers Konstantin Batygin1;2, Alessandro Morbidelli1, and Kleomenis Tsiganis3 1 Departement Cassiopee:´ Universite de Nice-Sophia Antipolis, Observatoire de la Coteˆ dAzur, 06304 Nice, France 2 Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125 3 Section of Astrophysics, Astronomy & Mechanics, Department of Phyiscs, Aristotle University of Thessaloniki, GR 54 124 Thessaloniki, Greece Preprint online version: October 29, 2018 ABSTRACT Context. The presence of highly eccentric extrasolar planets in binary stellar systems suggests that the Kozai effect has played an important role in shaping their dynamical architectures. However, the formation of planets in inclined binary systems poses a con- siderable theoretical challenge, as orbital excitation due to the Kozai resonance implies destructive, high-velocity collisions among planetesimals. Aims. To resolve the apparent difficulties posed by Kozai resonance, we seek to identify the primary physical processes responsible for inhibiting the action of Kozai cycles in protoplanetary disks. Subsequently, we seek to understand how newly-formed planetary systems transition to their observed, Kozai-dominated dynamical states. Methods. The main focus of this study is on understanding the important mechanisms at play. Thus, we rely primarily on analytical perturbation theory in our calculations. Where the analytical approach fails to suffice, we perform numerical N-body experiments. Results. We find that theoretical difficulties in planet formation arising from the presence of a distant (˜a ∼ 1000AU) companion star, posed by the Kozai effect and other secular perturbations, can be overcome by a proper account of gravitational interactions within the protoplanetary disk. In particular, fast apsidal recession induced by disk self-gravity tends to erase the Kozai effect, and ensure that the disk’s unwarped, rigid structure is maintained. Subsequently, once a planetary system has formed, the Kozai effect can continue to be wiped out as a result of apsidal precession, arising from planet-planet interactions. However, if such a system undergoes a dynamical instability, its architecture may change in such a way that the Kozai effect becomes operative. Conclusions. The results presented here suggest that planetary formation in highly inclined binary systems is not stalled by pertur- bations, arising from the stellar companion. Consequently, planet formation in binary stars is probably no different from that around single stars on a qualitative level. Furthermore, it is likely that systems where the Kozai effect operates, underwent a transient phase of dynamical instability in the past. Key words. Planets and satellites: formation – Planets and satellites: dynamical evolution and stability – Methods: analytical – Methods: numerical 1. Introduction i is the inclination), and libration of its argument of perihelion ! ◦ around ±90 . A necessary criterion forp the resonance is a suffi- Among the most unexpected discoveries brought forth by a con- ciently large inclination (i > arccos 3=5) relative to the mas- tinually growing collection of extra-solar planets has been the re- sive perturber’s orbital plane, during the part of the cycle where alization that giant planets can have near-parabolic orbits. Since the test-particle’s orbit is circular. the seminal discovery of 16Cygni B (Cochran et al., 1997), fol- lowed by HD80606 (Naef et al., 2001), much effort has been By direct analogy with the Sun-Jupiter-asteroid picture, the dedicated to understanding the dynamical origin and evolution Kozai resonance can give rise to variation in orbital eccentricity of systems with highly eccentric planets. In particular, it has and inclination of an extra-solar planet, whose orbit, at the time been understood that in presence of a companion star on an of formation, is inclined with respect to a stellar companion of inclined orbital plane, the most likely pathway to production the planet’s host star (Wu and Murray, 2003). In the systems arXiv:1106.4051v1 [astro-ph.EP] 20 Jun 2011 of such extreme planet eccentricities is via Kozai resonance mentioned above (16Cygni B, HD80606) the stellar compan- (Eggleton and Kiseleva-Eggleton, 2001). ions’ (e.g. 16Cygni A, HD80607) proper motion has been ver- The Kozai resonance was first discovered in the context of ified to be consistent with a binary solution. Other examples of orbital dynamics of highly-inclined asteroids forced by Jupiter, planets in binary stellar systems are now plentiful (e.g. γ Cephei and has been subsequently recognized as an important process (Hatzes et al., 2003), HD 196885 (Correia et al., 2008), etc) with in sculpting the asteroid belt (Kozai, 1962) as well as being binary separation spanning a wide range (˜a ∼ 10 − 1000 AU). the primary mechanism by which long-period comets become However, all planets whose eccentricities are expected to have Sun-grazing (Bailey et al., 1992; Thomas and Morbidelli, 1996). been excited by the Kozai resonance with the companion star Physically, the Kozai resonance corresponds to extensive excur- are in wide binaries. sions in eccentricity and inclination of a test particle forced by a If a Kozai cycle is characterized by a sufficiently small peri- massive perturber,p subject to conservation of the third Delaunay helion distance, the eccentricity of the planet may subsequently momentum H = 1 − e2 cos(i) (where e is the eccentricity and decay tidally, yielding a pathway to production of hot Jupiters, 2 Batygin et al.: Planet Formation in Inclined Binaries whose orbital angular momentum vector is mis-aligned with re- γ (arcsec/year) spect to the stellar rotation axis (Fabrycky and Tremaine, 2007). The presence of such objects has been confirmed via observa- 2 tions of the Rossiter-McLaughlin effect (McLaughlin, 1924), 0 leading to a notion that Kozai cycles with tidal friction are re- numerical sponsible for generating at least some misaligned systems (Winn -2 et al., 2010; Morton and Johnson, 2011). Kozai cycles may have also played an important role in - -44 systems where a stellar companion is not currently observed. Indeed, one can envision an evolutionary history where the bi- perihelion frequnecy -6 nary companion gets stripped away as the birth cluster disperses. In fact, such a scenario may be rather likely, as the majority of - -88 analytical stars are born in binary systems (Duquennoy and Mayor, 1991). -10 a (AU) 16 1820 22 24 2628 28 30 3232 In this case, a Kozai cycle can be suddenly interrupted, causing semi-major axis (AU) the planet’s eccentricity to become ”frozen-in.” In face of the observationally suggested importance of Kozai Fig. 1. An example of apsidal precession,γ, in a self-gravitating cycles during early epochs of planetary systems’ dynamical evo- disk. Here the disk is assumed to contain 50M⊕ between 16 and lution, the formation of planets in presence of a massive, inclined Sunday,32 May 29, AU, 2011 characteristic of a typical post-formation debris disk in perturber poses a significant theoretical challenge (Larwood et the Nice model of solar system formation (Tsiganis et al., 2005). al., 1996; Marzari et al., 2009; Thebault´ et al., 2010). After all, The solid curve shows the precession rate predicted by eqs. (1)- in the context of the restricted problem (where only the stars (3), as a function of semi major axis. The dots and error bars are treated as massive perturbers), one would expect the proto- show the results of a numerical calculation, integrating 3,000 planetary disk to undergo significant excursions in eccentricity equal-mass particles with a softening parameter of ≈ 0:005 AU and inclination due to the Kozai resonance, with different tem- to smooth the effects of their mutual close encounters. The disk poral phases at different radial distances, resulting in an inco- was binned into 100 annuli in a and the mean frequency of the herent structure. Such a disk would be characterized by high- longitude of pericenter$ ˙ was measured from the time-series of velocity impacts among newly-formed planetesimals, strongly $ of the particles in each bin (dots) as well as its variance (error inhibiting formation of more massive objects (planetary em- bars). Note that the precession frequency of a self-gravitating bryos) (Lissauer, 1993). disk is negative. Damping of eccentricities due to gas-drag has been consid- ered as an orbital stabilization process. However, excitation of mutual inclination among neighboring annuli renders this mech- alytic approach, that gravity is a sufficient mechanism to main- anism ineffective (Marzari et al., 2009). Ultimately, in the con- tain orbital coherence and planetary growth, without any need text of a restricted model, one is forced to resort to competing to account for other forces acting inside the disk. In fact, we time-scales for formation of planetesimals and dynamical ex- show that one of the primary effects of self-gravity is to induce citation by the companion star. Such an analysis suggests that a fast, rigid recession in the longitudes of perihelion and ascend- although possible, planetary formation in binary systems is an ing node of the disk. This allows for planetary formation to take unlikely event. place, as if the secular perturbations arising from the stellar com- Here, we show that the theoretical difficulties in planet for- panion were not present. It is noteworthy
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